Editing SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1

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=Virial Equilibrium of Adiabatic Spheres (1<sup>st</sup> Effort)=

<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="50%"><br />[[SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1|Part I: &nbsp; Isolated Configurations]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="50%"><br />[[SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt2|Part II:&nbsp; Configurations Embedded in an External Medium]]<br />&nbsp;</td>
</tr>
</table>

Highlights of the rather detailed discussion presented below have been [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|summarized in an accompanying chapter]] of this H_Book.

==Review==
===Adopted Normalizations===
In an [[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems.
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">
Adopted Normalizations for Adiabatic Systems
</th></tr>
<tr><th align="center">
In Terms of <math>~\gamma_g</math>
</th>
<th align="center">
In Terms of <math>~n</math>
</th></tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma_g} \biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K^4}{G^{3\gamma_g} M_\mathrm{tot}^{2\gamma_g}} \biggr]^{1/(4-3\gamma_g)}  </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ P_\mathrm{norm} R_\mathrm{norm}^3 = 
\biggl[ KG^{3(1-\gamma_g)}M_\mathrm{tot}^{6-5\gamma_g} \biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma_g )}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl[ \frac{K}{(G^3 M_\mathrm{tot}^2)^{\gamma_g-1}} \biggr]^{1/(4-3\gamma_g )}  </math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{n/(n-3 )}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{4\pi}{3} \biggl[ \frac{K^n}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(n-3 )}  </math>
  </td>
</tr>
</table>

</td>

</tr>
</table>
</div>

===Virial Equilibrium===
Also in our introductory discussion &#8212; see especially the section titled, [[SSCpt1/Virial#Energy_Extrema|''Energy Extrema'']] &#8212; we deduced that an adiabatic system's dimensionless equilibrium radius,
<div align="center">
<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, ,</math>
</div>
is given by the root(s) of the following equation:
<div align="center">
<math>
2\mathcal{C} \chi_\mathrm{eq}^{-2}  + ~3 \mathcal{B}\chi_\mathrm{eq}^{3 -3\gamma_g} -~3\mathcal{A}\chi_\mathrm{eq}^{-1}  
-~ 3\mathcal{D} \chi_\mathrm{eq}^3 = 0 \, ,
</math>
</div>
where the definitions of the various coefficients are,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{\gamma}
\cdot \mathfrak{f}_A 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} 
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} 
\cdot \mathfrak{f}_A \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
  </td>
</tr>
</table>
</div>

(The [[SSCpt1/Virial#Structural_Form_Factors|dimensionless structural form factors]], <math>\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[SSC/Virial/Polytropes#Role_of_Structural_Form_Factors|discussed further, below]].)  Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M_\mathrm{tot}</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) have been specified, the values of all of the coefficients are known and <math>\chi_\mathrm{eq}</math> can be determined.


<div align="center" id="ABratios">
<table border="1" cellpadding="10" align="center">
<tr><td align="left">

For later use, we note that after making the substitution, <math>~\gamma_g \rightarrow (n+1)/n</math>, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathcal{A}}{\mathcal{B}} \biggr)^n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{  
\frac{3}{20\pi} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)/n}
\cdot \mathfrak{f}_A^{-1}
\biggr\}^n</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5^{-n}
\biggl( \frac{3}{4\pi} \biggr)^n \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{2n} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]^{-(n+1)}
\cdot \biggl( \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr)^{n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5^n} \biggl( \frac{4\pi}{3} \biggr)  \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1}
\cdot \frac{\mathfrak{f}_W^n}{\mathfrak{f}_A^n \mathfrak{f}_M^{n-1}} \, ;
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathcal{B}^{4n}}{\mathcal{A}^{3(n+1)}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 
\frac{1}{5} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \biggr\}^{-3(n+1)}
\biggl\{ \frac{4\pi}{3}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n}
\cdot \mathfrak{f}_A
\biggr\}^{4n}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
5^{3(n+1)}  \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^{-6(n+1)} 
\biggl( \frac{4\pi}{3} \biggr)^{4n-4(n+1)}
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]^{4(n+1)} \cdot\frac{\mathfrak{f}_A^{4n}}{\mathfrak{f}_W^{3(n+1)}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
5^{3(n+1)} \biggl( \frac{3}{4\pi} \biggr)^{4} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2(n+1)} 
\cdot\frac{\mathfrak{f}_A^{4n} \mathfrak{f}_M^{2(n+1)} }{\mathfrak{f}_W^{3(n+1)}}
</math>
  </td>
</tr>
</table>
</div>

</td></tr>
</table>
</div>

==Isolated Nonrotating Adiabatic Configuration==
For a nonrotating configuration <math>~(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>~(D=P_e = 0)</math>, the statement of virial equilibrium is,
<div align="center">
<math>
3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1}  = 0 \, .
</math>
</div>

Hence, one equilibrium state exists for each value of <math>~\gamma_g</math> and it occurs where,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g} = \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_g} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{A}{B} \, .
</math>
  </td>
</tr>
</table>
</div>

<div align="center" id="TwoPointsOfView">
<table border="1" width="100%" cellpadding="8">
<tr>
  <th align="center" colspan="2">
Two Points of View
  </th>
</tr>

<tr>
  <td align="center" colspan="1">
In terms of <math>~K</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math>
  </td>
  <td align="center" colspan="1">
In terms of <math>~P_c</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 
\biggl\{ 
\frac{3}{4\pi} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{-{\gamma_g}}
\cdot \frac{1}{\mathfrak{f}_A}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~R_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3\cdot 5} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2-{\gamma_g}}
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggl[
\frac{GM_\mathrm{tot}^{2-\gamma_g}}{K} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~\frac{KR_\mathrm{eq}^{4-3\gamma_g}}{GM_\mathrm{limit}^{2-\gamma_g}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{5} \biggl( \frac{4\pi}{3} \biggr)^{\gamma_g-1} \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^{2-\gamma_g}}  
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
&#8212; &#8212; &#8212; &#8212; &#8212; &#8212; &nbsp; &nbsp; or, 
inverted and setting <math>~\gamma_g = 1 + 1/n</math> &nbsp; &nbsp; &#8212; &#8212; &#8212; &#8212; &#8212; &#8212;
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
<math>~
4\pi \biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} R_\mathrm{eq}^{3-n}
=
\biggl( \frac{5 \mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \frac{3}{\mathfrak{f}_M}
</math>
  </td>
</tr>

</table>

  </td>
  <td align="center" colspan="1">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\biggl\{ 
\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 
\biggl\{ 
\frac{3} {4\pi}
\biggl[ \biggl( \frac{P_\mathrm{norm}}{P_c} \biggr)\chi^{-3\gamma} \biggr]_\mathrm{eq} 
\cdot \frac{1}{\mathfrak{f}_A}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\chi_\mathrm{eq}^{4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\biggl( \frac{P_\mathrm{norm}}{P_c} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>
</table>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{P_\mathrm{norm} R_\mathrm{norm}^4} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr)^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{G M_\mathrm{limit}^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>
</div>

According to the solution shown in the left-hand column, for fluid with a given specify entropy content, the equilibrium mass-radius relationship for adiabatic configurations is,
<div align="center">
<math>
M_\mathrm{tot}^{(\gamma_g - 2)} \propto R_\mathrm{eq}^{(3\gamma_g -4)} \, .
</math>
</div>
We see that, for <math>~\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass.  Conversely, for <math>~\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius.  For <math>~\gamma_g > 2</math> or <math>~\gamma_g <  4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii.  However, for <math>~\gamma_g</math> in the range, <math>~2 > \gamma_g > 4/3</math>, configurations with larger mass have smaller equilibrium radii.  (Note that the related result for [[SSC/Virial/Isothermal#Virial_Equilibrium_of_Isothermal_Spheres|isothermal configurations]] can be obtained by setting <math>~\gamma_g = 1</math> in this adiabatic solution, because <math>~K = c_s^2</math> when  <math>~\gamma_g = 1</math>.)

==Role of Structural Form Factors==
When employing a virial analysis to determine the radius of an equilibrium configuration, it is customary to set the structural form factors, <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, to unity and accept that the expression derived for <math>~R_\mathrm{eq}</math> is an estimate of the configuration's radius that is good to within a factor of order unity.  As has been demonstrated in our [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model|related discussion of the equilibrium of uniform-density spheres]], these form factors can be evaluated if/when the internal structural profile of an equilibrium configuration is known from a complementary detailed force-balance analysis.  In the case being discussed here of isolated, spherical polytropes, [[SSC/Structure/Polytropes#Lane-Emden_Equation|solutions to the]],
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
can provide the desired internal structural information.  Here we draw on [[Appendix/References|Chandrasekhar's [C67]]] discussion of the structure of spherical polytropes to show precisely how our structural form factors can be expressed in terms of the Lane-Emden function, <math>~\Theta_H</math>, dimensionless radial coordinate, <math>~\xi</math>, and the function derivative, <math>~\Theta^' = d\Theta_H/d\xi</math>. 
 
===Mass===
We note, first, that [[Appendix/References|Chandrasekhar [C67]]] &#8212; see his Equation (78) on p. 99 &#8212; presents the following expression for the mean-to-central density ratio:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\bar\rho}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} \, ,</math>
  </td>
</tr>
</table>
</div>
where the notation at the bottom of the closing square bracket means that everything inside the square brackets should be, "evaluated at the surface of the configuration," that is, at the radial location, <math>~\xi_1</math>, where the Lane-Emden function, <math>~\Theta_H(\xi)</math>, first goes to zero.  But, as we pointed out when [[SSCpt1/Virial#Structural_Form_Factors|defining the structural form factors]], the form factor associated with the configuration mass, <math>~\mathfrak{f}_M</math>, is equivalent to the mean-to-central density ratio.  We conclude, therefore, that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} \, .</math>
  </td>
</tr>
</table>
</div>


===Gravitational Potential Energy===
Second, we note that [[Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy &#8212; see his Equation (90), p. 101 &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{5-n} \biggl( \frac{GM^2}{R} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
whereas our analogous expression is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, .</math>
  </td>
</tr>
</table>
</div>
We conclude, therefore, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{5-n} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} \, .</math>
  </td>
</tr>
</table>
</div>

===Mass-Radius Relationship===
Third, [[Appendix/References|Chandrasekhar [C67]]] shows &#8212; see his Equation (72), p. 98 &#8212; that the general mass-radius relationship for isolated spherical polytropes is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~GM^{(n-1)/n} R^{(3-n)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(n+1)K}{(4\pi)^{1/n}} \biggl[ - \xi^{(n+1)/(n-1)} \frac{d\Theta_H}{d\xi} \biggr]^{(n-1)/n}_{\xi=\xi_1} \, ,
</math>
  </td>
</tr>
</table>
</div>
which we choose to rewrite as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~(n+1)^n\biggl[ \xi^{(n+1)} (-\Theta^')^{(n-1)}\biggr]_{\xi=\xi_1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{\xi}{\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1} \, .
</math>
  </td>
</tr>
</table>
</div>

By comparison, the expression for the equilibrium radius that has been derived, above, from an analysis of extrema in the free energy function &#8212; specifically, see the last expression in the left-hand column of the [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]] &#8212; we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, it appears as though, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ \frac{1}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{\xi}{3\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1}  \, .
</math>
  </td>
</tr>
</table>
</div>
Or, taking into account the expressions for <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math> that have just been uncovered, we conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ (n+1) \xi ~(-\Theta^') \biggr]_{\xi=\xi_1}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{\mathfrak{f}_A}{\mathfrak{f}_W}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{(n+1) }{3\cdot 5} ~\xi_1^2 \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{(n+1) }{3\cdot 5} ~\xi_1^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1}  \biggr\}\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}  \, .
</math>
  </td>
</tr>
</table>
</div>

===Central and Mean Pressure===
It is also worth pointing out that [[Appendix/References|Chandrasekhar [C67]]] &#8212; see his Equations (80) &amp; (81), p. 99 &#8212; introduces a dimensionless structural form factor, <math>~W_n</math>, for the central pressure via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~W_n \biggl( \frac{GM^2}{R^4} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
and demonstrates that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{W_n}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~4\pi (n+1) \biggl[ \Theta^' \biggr]^2_{\xi_1} \, .</math>
  </td>
</tr>
</table>
</div>
It is therefore clear that a spherical polytrope's central pressure is expressible in terms of our structural form factor, <math>~\mathfrak{f}_A</math>, as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{4\pi (5-n)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) \frac{1}{\mathfrak{f}_A} \, .</math>
  </td>
</tr>
</table>
</div>

Looking back at our [[SSCpt1/Virial#FormFactors|original definition of the structural form factors]], we note that,
<div align="center">
<math>\mathfrak{f}_A = \biggl( \frac{\bar{P}}{P_c} \biggr)_\mathrm{eq} \, .</math>
</div>
Hence, this last equilibrium relation can be rewritten as,
<div align="center">
<math> \frac{\bar{P} R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} = \frac{3}{4\pi (5-n)} \, .</math>
</div>

===Alternate Derivation of Gravitational Potential Energy===

<!-- BEGINNING OF DELETION:  The following subsubsection is redundant, now that the "Two Points of View" table has been included.

====Central Pressure====
As is pointed out, above, in our [[SSC/Virial/Polytropes#Mass-Radius_Relationship|discussion of the mass-radius relationship]], the expression for the equilibrium radius that has been derived from our analysis of extrema in the free energy function can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, .
</math>
  </td>
</tr>
</table>
</div>
Via the polytropic equation of state, we can relate <math>~K</math> to the central pressure as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~K \rho_c^{1+1/n} = K \bar\rho^{1+1/n} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{1+1/n}
= K \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)/n} \biggl( \frac{1}{\mathfrak{f}_M} \biggr)^{(n+1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ K^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
P_c^{n} \biggl[ \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr]^{(n+1)} \mathfrak{f}_M^{(n+1)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, in the mass-radius relationship we can replace <math>~K</math> with this expression to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{P_c}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)} 
\biggl[ \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr]^{(n+1)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~3 \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \biggl( \frac{G M_\mathrm{tot}^2}{P_c R_\mathrm{eq}^4}\biggr)^n 
\biggl( \frac{3}{4\pi } \biggr)^{n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M^2}{\mathfrak{f}_W} \biggr)^n 
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
<math>
\Rightarrow ~~~~ P_c = \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2}  \, .
</math>
  </td>
</tr>
</table>
</div>
Now, from our above [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|examination of the expression for the gravitational potential energy]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{5}{5-n} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, our expression for the central pressure becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="3">
<math>
~P_c = \frac{3 }{4\pi (5-n)}\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{1}{\mathfrak{f}_A }  \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, this precisely matches [[SSC/Virial/Polytropes#Central_Pressure|the expression that we derived, above]], starting from [[Appendix/References|Chandrasekhar's [C67]]] presentation.
====Gravitational Potential Energy====


END OF DELETION  -->

As has been [[SSCpt1/Virial#AlternateGravPotEnergy|discussed elsewhere]], we have learned from [[Appendix/References|Chandrasekhar's discussion of polytropic spheres [C67]]] &#8212; see his Equation (16), p. 64 &#8212; that if a spherically symmetric system is in hydrostatic balance, the total gravitational potential energy can be obtained from the following integral:
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ + \frac{1}{2} \int_0^R \Phi(r) dm \, .</math>
  </td>
</tr>
</table>
</div>
Using "[[SSCpt2/SolutionStrategies#Technique_3|technique #3]]" to solve the differential equation that governs the statement of hydrostatic balance, we know that in any polytropic sphere, <math>~\Phi(r)</math> is related to the configuration's radial enthalpy profile, <math>~H(r)</math>, via the algebraic expression,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi(r) + H(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C_B \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~C_B</math>, is an integration constant.  At the surface of the equilibrium configuration, <math>~H = 0</math> and <math>~\Phi = - GM_\mathrm{tot}/R_\mathrm{eq}</math>, so the integration constant is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~C_B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math>
  </td>
</tr>
</table>
</div>
which implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi(r) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - H(r) - \frac{GM_\mathrm{tot}}{R_\mathrm{eq}}  \, .</math>
  </td>
</tr>
</table>
</div>
Now, from [[SR#Barotropic_Structure|our general discussion of barotropic relations]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~H(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n+1) \frac{P(r)}{\rho(r)} \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-\Phi(r)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{1}{2} \int_0^R \biggl[ (n+1) \frac{P(r)}{\rho(r)} + \frac{GM_\mathrm{tot}}{R_\mathrm{eq}} \biggr] 4\pi \rho(r) r^2 dr </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
(n+1) \int_0^R P(r) r^2 dr 
+ \frac{GM_\mathrm{tot}}{R_\mathrm{eq}}  \int_0^R  \rho(r) r^2 dr 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
\frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \int_0^1 3\biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx 
+ \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}}  \biggl( \rho_c R_\mathrm{eq}^3  \biggr) \int_0^1  3 \biggl[ \frac{\rho(x)}{\rho_c} \biggr] x^2 dx 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\pi \biggl\{ 
\frac{1}{3} (n+1)P_c R_\mathrm{eq}^3 \mathfrak{f}_A 
+ \frac{GM_\mathrm{tot}}{3R_\mathrm{eq}}  \biggl( \rho_c R_\mathrm{eq}^3  \biggr) \mathfrak{f}_M 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ 
\frac{2\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A 
+  \frac{1}{2} \biggl[ \frac{4\pi \rho_c R_\mathrm{eq}^3}{3M_\mathrm{tot}}  \biggr] \mathfrak{f}_M 
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{1}{2} \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggl\{ 
\frac{4\pi}{3} (n+1) \biggl[ \frac{P_c R_\mathrm{eq}^4}{GM_\mathrm{tot}^2} \biggr] \mathfrak{f}_A 
+  1 
\biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
We now recall two earlier expressions that show the role that our structural form factors play in the evaluation of <math>~W</math> and <math>~P_c</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~P_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3 }{20\pi }\biggl( \frac{G M_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \, .</math>
  </td>
</tr>
</table>
</div>
Plugging these into our newly derived expression for the gravitational potential energy gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~- \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{1}{2} \biggl\{ 
\frac{4\pi}{3} (n+1) \biggl[ \frac{3 }{20\pi }\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^2} \biggr] \mathfrak{f}_A 
+  1 \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ (2\cdot 3) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
(n+1) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} +  5 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ (5 - n) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  5 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  \frac{5}{5-n} \, . </math>
  </td>
</tr>
</table>
</div>
As it should, this agrees with the expression for the ratio, <math>\mathfrak{f}_W/\mathfrak{f}_M^2</math>, that was [[SSC/Virial/Polytropes#Gravitational_Potential_Energy|derived in our above discussion of the gravitational potential energy]].

===Summary===
In summary, expressions for the three structural form factors associated with isolated, spherically symmetric polytropes are as follows:

<div align="center">
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Structural Form Factors for <font color="red">Isolated</font> Polytropes
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<math>~\mathfrak{f}_M</math>
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<math>~=</math>
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<math>~ \biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\xi_1} </math>
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<math>\mathfrak{f}_W </math>
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<math>~=</math>
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<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1} </math>
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<math>\mathfrak{f}_A  </math>
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<math>~=</math>
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<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}  
</math>
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=See Also=


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